This chapter explains how to conduct and interpret analyses of associations between two categorical variables. By the end of this chapter, you will be able to:

• Understand and explain the concepts of the chi-square test.

• Evaluate the assumptions of the chi-square test.

• Understand the results from the chi-square test, as well as the power of the effect.

• Distinguish between relative risk and odds ratio.

• Report the findings in APA style correctly.

• Understand how and when to use associations of categorical variables in evidence-based practice.

• Categorical data

• Chi-square test

• Contingency tables

• Crosstab analysis

• Fisher's exact test

• Odds ratio

• Phi and Cramer's V

• Relative risk

Nurses are often interested in relationships between variables. Such investigations are important steps in the development of interventions that are used to improve patient outcomes. When variables are measured at the interval or ratio level (also called continuous variables), we use Pearson's r to examine the relationship between variables. However, some variables of interest are categorical in nature (i.e., nominal or ordinal), such as patient outcomes (improved/not improved), diagnoses (presence/absence of a condition), or demographic categories (race, gender identity).

For example, we might be interested in variables such as blood type (A, B, O, or AB), response level to treatment (recovery, disability, or death), or ethnicity. These data are measured at the nominal or ordinal level and do not lend themselves to arithmetic analyses. These types of data are often called categorical data. A categorical variable may be examined through a frequency analysis (e.g., counting the number of times the patient recovers) as we saw in Chapter 6. Still, a frequency analysis does not help us answer a more sophisticated question, such as understanding the relationship between two or more categorical variables. Crosstab analysis is a simple extension of frequency analysis that allows us to examine the association between two or more categorical variables in terms of joint frequencies. As both variables are categorical, descriptive mean and standard deviation statistics do not make sense. Instead, we analyze the frequency and/or the percentage/proportion to answer questions about associations between variables.

Investigators may examine the associations between categorical variables to understand health outcomes and diseases in relation to socioeconomic factors, race and ethnicity, geographical location, and so on. An investigator might ask a question such as, "Is there a relationship between where a patient resides (urban or rural) and their health status (excellent, good, fair, or poor)?" In this case, a variable, place of residence, is measured at the nominal level and has two possible choices; the other variable is ordinal and has four possible choices. Similarly, the nurse investigator might ask a quality improvement question such as, "What is the relationship between the presence or absence of a patient navigator (a nurse who helps patients negotiate complex health problems and treatment) and patient satisfaction (high, moderate, or low)?"

Interpreting the results of these types of studies must be made with an understanding of each approach's strengths and limitations and the statistical analysis's underlying assumptions. As a nurse investigator, you must recognize what variables are being studied, what hypotheses are proposed, and the level of measurement for each variable to choose the most appropriate statistical test for the analysis.

The chi-square test is the simplest method to analyze variables that are measured at the categorical level. The chi-square test is a type of nonparametric test that does not make a distributional assumption, making it suitable for a wide range of healthcare data. In the analysis, both Excel and IBM SPSS Statistics software (SPSS) create a crosstab between the two variables and then analyze the frequencies for a possible association. The following are example questions that can be addressed with a chi-square test:

• Is there an association between infant mortality and income (e.g., low, middle, high)?

• Is the type of wound care dressing (standard gauze, hydrocolloid, or foam) related to wound healing outcome (healed vs. not healed) among postoperative patients?

• Is there an association between a patient's smoking status (person who smokes vs. who does not) and the presence of respiratory complications (yes vs. no) following surgery?

• Is satisfaction level related to types of health insurance coverage?

A contingency table presents the frequency distribution of the two categorical variables, and the chi-square test can be used to test the relationship between variables for any contingency table that is 2 × 2 (i.e., both variables are dichotomous at either the nominal or ordinal level of measurement) or larger. An example of a 2 × 2 table is presented in Table 15-1. Note that in this example, there are two variables, age and antihistamine use, and each has two choices: for age, older than 30 years or younger than 30 years, and for antihistamine use, use or do not use. The 2 × 2 refers to the number of categories of variables being analyzed simultaneously (i.e., both variables are dichotomous). A 3 × 3 table might be an analysis of health status (poor, good, and excellent) by income (low, middle, and high).

One of the assumptions of a chi-square test was that the expected count in all cells should be greater than 1 and no more than 20% of cells should have a count of less than 5. The chi-square test becomes unstable (a less reliable and less valid analytic tool) when this assumption is violated (i.e., expected count in any cell is small). Fisher (1922), the famous statistician, developed Fisher's exact test to resolve this problem. Note that Fisher's exact test should only be used with 2 × 2 contingency tables when the sample size is small enough that the assumption can be violated. As we have seen in Table 15-3, the output of the chi-square test reports additional statistics, including Fisher's exact test.

Let us consider the following output where we have slightly modified the data so that some cells have an expected frequency of less than 5. That output is presented in Table 15-4. Notice that the cells for people younger than 30 who do not use antihistamines and those younger than 30 who use antihistamines have expected frequencies of less than 5, which violates the assumption. Therefore, the result of Fisher's exact test should be reported instead of that of the chi-square test. In our example, both p-values are small enough to rule out chance, so the results indicate the association between the age group and whether they use antihistamines.

When "Chi-square" is checked in the Crosstabs Statistics dialogue box, it calculates the statistic with its associated p-value. It indicates whether the two categorical variables are related. However, the small p-value associated with the chi-square statistic only indicates that the variables are related; it does not reveal anything about the strength of the relationship. In the Crosstabs Statistics box, there are two statistics that we can calculate and report when the strength of association is of interest; they are Phi and Cramer's V. Click on the "Statistics" button and check "Phi and Cramer's V," as displayed in Figure 15-9, to calculate measures for the strength of association between the two variables. Clicking "Continue" and then "OK" will then produce the output. The example output is presented in Table 15-5; note that this output is added to Table 15-3.

A screenshot in S P S S shows two columns of numerical data. A dialog box, with heading Crosstabs Statistics, shows several options listed; the option Phi and Cramer’s V is checked.

Reprint Courtesy of International Business Machines Corporation, © International Business Machines Corporation. "IBM SPSS Statistics software ("SPSS")". IBM^®, the IBM logo, ibm.com, and SPSS are trademarks or registered trademarks of International Business Machines Corporation.

Phi is used when both variables are dichotomous (when there are only two outcomes such as "yes" or "no," "present" or "not present"), but Cramer's V should be used when one of the variables is measured for more than two categories. Phi (φ) ranges between −1 and 1, and Cramer's V ranges between 0 and 1; the general rule for correlation coefficients can be used to interpret them (as discussed in Chapter 10). As the coefficient approaches 0, the association between the two categorical variables is weak but stronger as it approaches 1. Negative and positive signs will indicate whether the association is a positive or negative relationship. Note that these statistical tests can also be used to measure effect size. The reporting of these statistics can be added after the associated p-value with chi-square test statistics. Here is a sample report:

• There was a weak association between the age group and antihistamine use, x² (1) = 5.00, p = .025, ϕ = −.15.

The chi-square test is the preferred method to investigate an association between two categorical variables. Assumptions include independence of observations and adequate sample size for each cell. Because it has no distributional assumption, it is a nonparametric test.

In cases where any cell contains a frequency of less than 5 in 2 × 2 contingency tables, Fisher's exact test can be used and reported instead of a chi-square test because the chi-square test becomes unstable with a small sample size.

The small p-value associated with the chi-square statistic only indicates that the variables are related, so we may need a different measure of the strength of association. Phi and Cramer's V are such measures. Phi is reported for 2 × 2 contingency tables, whereas Cramer's V is reported for larger contingency tables.

RRs and ORs can also be computed and reported as effect sizes. RR is defined as the measure of the risk of an outcome (e.g., disease or death) occurring when exposed to a risk factor, and OR is defined as the ratio of an outcome (e.g., disease or death) occurring when exposed to a risk factor. Both should be reported with their associated CI.

1. Explain why the chi-square test is part of the family of nonparametric tests.

2. An investigator found that there is an association between gender identity and pain level (low versus high) with a small p-value, and the associated Phi measure was 30. How would the researcher interpret this measure? Explain.

3. Briefly describe a potential study in your field where you might use a chi-square test.

   1. Provide a one- or two-sentence description of the study (e.g., nature of the subjects/participants, purpose).

   2. State the research question that a chi-square test would answer.

   3. Describe the variables to be used for the chi-square test.

   4. Indicate what a low p-value associated with the chi-square test statistic in this study would state.

4. If a new intervention for pressure ulcer prevention yields an RR = 0.5, what does that mean for patients receiving the intervention? How might this be communicated to clinicians?

5. How would you decide between using a chi-square test of independence versus a logistic regression model for binary outcomes in a clinical study?

6. Describe a hypothetical nursing scenario using Fisher's exact test rather than the chi-square test.

7. In a nursing study, a statistically significant chi square was found for "smoking status" versus "postoperative complications," but Phi = 0.08. What might you conclude?

8. If the author of a journal article reports a statistically significant chi-square value but does not provide any measure of association (like Phi or Cramer's V) or effect size, what additional questions might you ask or what additional information would you seek?

9. In a case-control study about a rare infection, an odds ratio (OR) = 3 is found for patients exposed to a certain risk factor. How would you explain this finding to a colleague who is not statistically trained?

10. Why might a researcher choose the chi-square test of independence instead of a t-test when comparing two groups?

1. True or False: A chi-square test result with a small p-value will indicate that the two variables are associated and how strong a relationship is.

2. Which of the following statistics should be reported when any cell has an expected frequency of less than 5 in a 2 × 2 contingency table?

   1. Chi-square

   2. Likelihood ratio

   3. Fisher's exact test

   4. Linear-by-linear association

3. True or False: Cramer's V is a more appropriate measure for the strength of association than Phi when the size of a contingency table is 2 × 2.

4. True or False: RR is used with a cohort study, whereas OR is used for a case-control study.

5. Phi (ϕ) is typically used to measure effect size for:

   1. any size contingency table.

   2. only 2 × 3 contingency tables.

   3. specifically 2 × 2 contingency tables.

   4. ordinal data analysis.

6. Odds ratio (OR) is best described as:

   1. the probability of an event occurring in a population.

   2. the ratio of probabilities of an event in two groups.

   3. the ratio of odds of an event in two groups.

   4. a measure exclusively used in randomized controlled trials.

7. True or False: A Cramer's V of 0.65 indicates an extremely weak relationship.

8. True or False: Both RR and OR compare the frequency of an outcome in exposed versus nonexposed groups, but RR directly compares risks while OR compares odds.

9. In a case control study of patients with a rare infection (cases) and without it (controls), researchers want to see if prior exposure to a specific contaminated source is associated with the infection. Which measure of association is most appropriate?

   1. Relative risk

   2. Odd ratio

   3. Phi coefficient

   4. Cramer's V

10. True or False: A p-value from a chi-square test of independence alone indicates whether the association is statistically significant but does not inform you of the magnitude of that association.